Birational geometry of the moduli space of pure sheaves on quadric surface (Q1677882)

The authors study the birational geometry of the moduli space \(\mathbf{M}\) parametrizing stable sheaves on a smooth quadric surface with \(c_1 = (2,3)\) and Hilbert polynomial \(5m+1\). They describe an explicit birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow up and blow downs. They provide a geometric descriptions of all intermediate spaces and maps. As a consequence, they recover results of Maican computing the Poincaré polynomial of \(\mathbf{M}\) and showing the rationality of \(\mathbf{M}\).